The word "romance" seems to come from Latin, but no similar Latin word appears to mean anything related. Is there a good Latin word for a romance, a kind of an intimate relationship? I cannot think of anything close than amicitia, but that is not quite a romance unless I am mistaken.
I would suggest amor and especially its plural: amores.
I think the plural would be better because singular amor is quite common and generic, though it covers what we mean by "romance." My sense (despite the fact that L&S does not distinguish the meaning of singular and plural) is that amores has a slightly difference nuance, even though it still be used generically for more things than English "romance," e.g. the actual object of my romance.
A good example comes from Plautus' Mercator, which begins with Charinus laying out the argumentum:
Duas res simul nunc agere decretumst mihi:
et argumentum et meos amores eloquar.
Riley translates as:
Two things have I now resolved to do at the same time;
both the subject and my own amours will I disclose.
I think we can feasibly translate "meos amores eloquar" as: "I shall tell you of my romance."