Galician-Portuguese had its version of the old Roman names for weekdays, and we find a handful of instances of them in the 13th, 14th and 15th centuries in this Corpus do Português and Corpo Informatizado do Português Medieval (CIPM). Their numbers are easily dwarfed though by the Christian names segunda feira, terça feira, etc. in these corpora. The names are as follows:
Monday: lũes, lues, lunes
Thursday: joves (often spelled joues or ioues)
Friday: vernes (often spelled uernes)
The old Roman names for Sunday and Saturday had been completely replaced by the Latin Christian names, which by this time had already become domingo and sábado in Galician-Portuguese.
Wikipédia (Dias da Semana) states that back in those days these were the names in use in Galicia whereas segunda feira, etc. were used in Portugal. (In modern Galician the old names, with luns for lũes, and xoves for joves, are used alongside segunda feira, etc.) My findings are consistent with this, as most instances of the old Roman names that I found either come from Galicia or have some plausible Galician or Castilian connection. So maybe these names were not in use in Portugal by the 13th century anymore. In fact if Galicia used the old Roman names it is surprising that we do not find them more often, but maybe CIPM and Corpus do Português don’t have much material from Galicia.
Let’s see the examples I found one by one―they’re not that many―going from definitely Galician to probably having some Galician connection to I-have-no-idea-why-the-author-used-the-old-name.
Two examples are from 1400s legal documents from the region of Lugo in Galicia (my boldface in all quotes):
Feyto et outorgado […] enna çiudade de Lugo, martes, quatorse dias do mes de Setenbro, anno do nasçemëto do nosso Señor Jhesu Christo de mjll et quatroçëtos et quatorse annos. 
Que foron feytas & outorgadas enno dito mosteyro de Chantada, dja lues, a [21-03-1474].
Then we have three cases in the 1200s Cantigas de Santa Maria (Wikipedia). Spelling in these songs suggest a Galician origin, and several scholar think many of them were written by the Galician Nuno Airas and King Alfonso X of Castile, who lived in Galicia in his childhood, and quite liked Galician-Portuguese for poetry. There’s also a “sesta feira” in the songs.
[…] e chegaron ao Porto mercores, primeiro dia / d’abril, e ena ygreja entraron con gran conorte,
E chegou vernes aa ssa ygreja
E depus isto, vernes madrugada, / levava vinn’ e pan aa pousada
Then we have a 1300s Portuguese translation of the Siete Partidas de Alfonso X (Wikipedia):
E por ende a sancta igreia buscou maneira per que a crisma ffosse asy feyta en aquesta quinta feyra que chamã ioues de çea
E por quães rrazões deue seer onrrada e aguardada a festa de joues de quinta feyra de çena en que ha de seer sagrada a crisma.
The last quote is a translation of “[…] la fiesta del Jueves de la cena” (see this late Spanish edition), so it looks as though the translator adapted Jueves to joves, but kept it as the name of that festivity or particular Thursday, rather that the name for ‘Thursday’ in general, as he goes on to write “quinta feyra”.
Then we have a 1306 letter to or about a Dom João de Portel, written in “Merida.” I don’t know of any Mérida in Portugal, and this Mérida appears to be important enough to be the seat of a “cabidoo” (from Latin capitulus, ‘chapter; assembly of the clergy of a cathedral, church, or an order). Médida, Spain, is about 200 km from Portel.
[…] Ffeyta en Merida, en nosso Cabidoo geral, Martes, XX dias andados de Março, da era de mil e CCC e sex anos.
Down my list is a late 1200s cantiga de escárnio e maldizer (‘song of scorn’). You can get help understanding it here. The author, Pero Mendes da Fonseca, is Portuguese, from Viseu district, but he may have written the song in Castille. The song is about a guy taking some official position at the Military Order of Saint James of the Sword at Uclês, Cuenca, in Castille (see here). And “e do lũes ao martes” fits the metric and rhyme:
Chegou Paio de más artes
e nom leeu el nas partes
e do lũes ao martes
foi comendador d'Ocrês.
Now it gets trickier. The next examples come from 1489 religious book, Tratado de Confissom. The book is packed full of segundas feiras, terças feiras, etc., it being about what you can and cannot eat on each day of the week when you sleep around or do any other things you’re supposed not to do. But in one instance it has mercores and vernes instead. The only possible Galician connection I can see is that it was printed in Chaves, which lies only about 11 km from the Galician border.
iaiũe os mercores a uiãda de coresma e os uernes a pã e agoa
I have no remotely plausible Galician connection for the last two examples. One is the A Demanda do Santo Graal a 15th century manuscript (here a 1887 printed edition). We find “lunes”, but also “sesta feira”
Entam pedio suas armas e depois que foe armado, sobio em sseu cauallo, e encomendarom os monges a deos e andarom aquelle dia e outro ssem auentura achar. Assi que hũu dia lunes lhes aueeo demanhãa que chegarom a hũa cruz que sse partira em duas carreiras.
The last exemple is from Garcia Resende’s Cancioneiro Geral, a 1516 compilation of 15th and 16th century poetry:
Mas aqueste dia martes foy jnfeles pera mym o meu sangue me deu fim & rrompeo meus estendartes.
I left out the early 1200s Foros de Castelo Rodrigo. It is full of the old Roman names, but it is written in a hybrid of Galician-Portuguese and Leonese. Castelo Rodrigo in now part of Portugal, but belonged to the kingdom of León at the time of writing. Lindley Cintra studied the Foros and suggested they were written by a Galician trying to write Leonese. More information in Spanish here and in French here.
Galician-Portuguese had lũes, martes, mercores, joves, and vernes, but it looks as though they were used mostly in Galicia. It is possible that they were not much used or even well known in Portugal. The earliest Portuguese dictionary I know of, the 1570 Jerónimo Cardoso’s Dictionarium Latino-Lusitanicum & vice versa is consistent with this idea: it translates the old Roman names simply as segunda feira, etc.
Jerónimo Cardoso, Dictionarium Latino-Lusitanicum & vice versa, 1570