In general, it's a debate, whether we should mention the holiday (מעין המאורע) in the three faceted blessing or not. Rambam in Mishneh Torah Berakhot 3:13 says yes, and he seems to base his ruling on Yerushalmi Berakhot 6:1. However, Tosafot on Berakhot 44a never heard about this practice in their land, and were perplexed to read it from Rambam. Still, the Tur Orach Chayim 208 agrees with Rambam (Beit Yoseif mentions the debate with sources), and it is how it is codified in the Shulchan Arukh Orach Chayim 208:12.
Regarding your question in particular, whether we should say it on the intermediate days, I have found a decision taking it for granted, but it's not evident. Citing Yehudah ben Yitzchak the Or Zarua implies that they had a different text of the Rambam, which included the intermediate days as well (Hilkhot Seuda 181):
כתב בספרי הרמב"ם זצ"ל שיש להזכיר בה מעין המאורע בשבתות וימים טובים ובחוה"מ ובראשי חדשים.
The Rambam זצ"ל wrote in his books that one has to mention in it the reference to the holiday on the Shabbat, holidays, the intermediate days of the festival and new moons.
The following reasoning is the same as in the Tosafot. Still, another source, the Kaf haChayim to Orach Chayim 208:12 clearly rules to say it. CYLOR to know your local custom.