Interesting guesses in the responses here. It never occurred to me that this organization might be anything other than "the next iteration of the Death Eaters".

I suspect he's referring to other schools which he believes to be more capable of teaching Battle Magic than cursed Hogwarts.

Salem Witches' Institute or a dojo in Asia seem the obvious choices, so we can probably safely eliminate them (because when was the last time EY used the obvious solution in MOR?)

Earlier in the story, Quirrell gave Harry a Portkey claiming it would take him somewhere that he would be contacted by people who would take him to a magic school in America...

It is also possible that Quirrell is referring to narrower training in Battle Magic. He's made previous references to knowing assassins in other countries (whom he could put in touch with Harry), and this could be an extension of that.

I'm unsure if this cleanly fits - but if by training Q means in rationality, "certain organizations beyond Britain's shores" could be referring to CFAR. A case for this not fitting is of course that CFAR does not exist in 1992.

I can't now think of other (real, this world) organizations which fit the phrases of training, old enough, beyond Britain's shores, letters of recommendation, and be something EY would reference.

Otherwise it would be a gesture to an implied larger magical world. But that gesture itself seems...unnecessary, beyond implying that Q considers the future education of his students important.