While "Martha" is an English language name, I have never encountered a "Martha" in Ireland, and according to the CSO (Central Statistics Office), no variation of "Martha" has made it to the top 100 popular names for girls in the last 18 years. April, May and June are also used as girls names in other English speaking countries, but they aren't common in Ireland - Maya makes the list, but it isn't pronounced the same way as the month, and I don't think it's supposed to be derived from the month. Julie and Julia do make the list, and are probably derived from July (or are considered the female version of Julius that July got it's name from). (There are a few Averil/Avril/Aprils around, though).
The short answer is that no, Irish people don't usually use months for names, except for Nollaig, which is the Irish version of Noel, and the word for December. Though I notice that Noel hasn't made the list for boys names in the last 18 years, suggesting that it may have fallen out of favour.