How to specify a lover's gender

In English, the female equivalent of gay is lesbian. Well, it depends. If you're speaking informally then gay/lesbian could be both used. If speaking formally, you can use homosexual

You don't need any special vocabulary here. You can simply say:

A doctor’s daughter hides her female lover ...

This is perfectly fine and idiomatic and is probably the closest to the meaning of the sentence in the Original Poster's language. One could easily also say:

• A doctor’s daughter hides her gay/lesbian lover ...

However, this overtly draws attention to the lover's sexual orientation in a way that the more neutral rendering with female does not.

It would seem to me that based on your comment about another language that what you're looking for is a way to communicate the gender without creating a sentence that makes it transparent that the gender is somehow important or that the writer is purposefully attempting to communicate it.

I indeed agree with others that “female lover” would be the simplest answer, but like all the other answers in English that sentence implies that the lover's gender is relevant to the story, much like “rich lover” would.

There is no way to communicate the gender in English without that implication, as far as I know, as one can with words such as “brother” or “aunt”, which communicate gender without arousing that impression.

A doctor’s daughter hides her lover

Two suggestions

1. Use the lover's name, e.g.

A doctor’s daughter hides her lover, Janet ...

2. Ask on Writing Stack Exchange where people are constantly dealing with the difficulties of writing about gender these days

P.S. You should avoid asking exactly the same question on two different Stack Exchanges, so you could perhaps delete this one, or phrase the new version more generally.