Since this year Rosh Chodesh is on Friday which leads us into the shloshes yemai Hagbala and we are already allowed to cut hair, may one cut his hair on thursday since it is forbidden on Rosh Chodesh?
No, it would not be permitted to take a haircut on Thursday.
Although some permit taking a haircut on Friday, in honor of Shabbos (Peri Megadim, Eishel Avraham 493:5), this does not extend to Thursday. Those who do not take a haircut on Friday, because of the tzava’ah of Rabbi Yehuda Ha-chassid (many are not particular about this custom), will have to wait until Sunday.
In fact, there are poskim who disagree with the Peri Megadim, and rule that one would not be permitted to take a haircut even on Friday. See Rabbi Elimelech Bluth, LeTorah VeHoraah vol. 10, p.14, quoting from Rav Moshe Feinstein, who makes several distinctions between the Friday before Lag Ba’Omer and the Friday before the commencement of the yemei hagbalah.
However, even if we permit haircuts on Friday, the heter does not extend to Thursday.