I do not think it suggests that the ancient Israelites used a strictly lunar calendar for a number of reasons:
There is no evidence that the Gezer calendar is meant to be a calendar in the sense of setting forth a normative year. It is a description or proscription of the farming that would take place over a year, broken down by months. It might very well be a farming plan for a specific farm for a specific year.
A typical year in the Jewish calendar is 12 lunar months. And leap months
in biblical and second temple times were added every few years but were not added until the end of the year, so any prospective statement would presume 12 months.
The word for month ירח comes from the hebrew word for "moon" (just like the english word "month") but came to mean month in general, not just a lunar month. Therefore, the mere fact that the term used is ירח does not mean that it was strictly lunar at that time.
I am editing this answer because I have come across a Baraisa quoted in Baba Metzia 106b that is astonishingly similar to Gezer calendar. The Baraisa quotes a three-way dispute about the agricultural seasons:
רשב"ג משום ר"מ אומר וכן היה רבי שמעון בן מנסיא אומר כדבריו חצי תשרי מרחשון וחצי כסליו זרע חצי כסליו טבת וחצי שבט חורף חצי שבט אדר וחצי ניסן קור חצי ניסן אייר וחצי סיון קציר חצי סיון תמוז וחצי אב קיץ חצי אב אלול וחצי תשרי חום רבי יהודה מונה מתשרי ר"ש מונה ממרחשון
Rabbi Shimon ben Gamliel from Rabbi Meir said, and so to did Rabbi Shimon ben Menasia say like his words: Half of Tishrei, Marchesvan, and half of Kislev is the planting season; half of Kislev, Teves, and half of Shevat is the winter season (but the Gemara explains that this is the late planting season for barley and legumes); half of Shevat, Adar, and half of Nissan is the cold season; half of Nissan, Iyar, and half of Sivan is the harvesting season; half of Sivan, Tammuz, and half of Av is the summer season, and half of Av, Elul, and half of Tishre is the hot season. Rabbi Yehuda counts (the agricultural cycle) starting in Tishrei. Rabbi Shimon counts (the agricultrual cycle) staring in Marcheshvan.
To me this makes clear that the Gezer calendar--just like the Baraisa--is an agricultural calendar setting forth the ideal times for planting, harvesting, etc. in Eretz Yisrael.