IIRC, it's one of those urban myths that's been floating around. I remember first seeing it when the MACOs were introduced on Enterprise and the Gene's Vision crowd called it against, well, Gene's Vision to which others responded that the earliest ideas for TOS called for a Marine Detachment on board, though I don't recall if they ever provided a source. I've heard older fans say that there was a belief back in the day that the security officers were Marines, based on the belief of said fans who were under the impression that Marines always handle security aboard naval vessels, but that's just fan conjecture and not anything official.I know you love the idea of Starfleet Marines, but source?
As far as I know, the only time there was anything official on the subject of Starfleet Marines was in TFF, where behind the scenes sources do indicate those were Marines on the Nimbus III raid, though they were never identified as such onscreen. Also when Enterprise was in the early development stages, the NX-01's security personnel were supposed to be Marines and Malcolm Reed a Major, though that obviously changed when the show went into production. Curiously, when the MACOs were introduced, they pretty much took over security anyway, with Reed being the only Starfleet Security officer on the ship in the third and fourth seasons.
I don't count Colonel West as proof of anything since he was obviously not meant to be anything more than an Oliver North pastiche. I just go with the (now defunct) Litverse continuity's take on him that he was actually a Starfleet Admiral who everyone called Colonel as a nickname. At the very least, if he were actually a Colonel, then he should have had a Captain's insignia on his uniform instead of an Admiral's.