Can we use 'for' and 'since' in a single sentence?

There is no rule against using "for" and "since in the same sentence, for example:

He has been looking for his father ever since he went missing.

The reason your sentence is a little jarring is that you are using "for" and "since" for exactly the same purpose - to mark the time something began. So you have a redundancy - saying the same thing twice.

Also, as "furlough" is a period of leave, I'm not sure "on frequent furlough" is correct. I think it should be "taking frequent furloughs" (if you mean multiple instances) or simply "on furlough".

You could simply say:

He has been taking frequent furloughs for the last four months.

This tells you how long he has been on "frequent furlough" - four months until now.

He has been taking frequent furloughs since his father's death.

This also tells you how long he has been on furlough - presuming you know the date of his father's death.

If you wanted to include the fact that his father died and that it was four months ago, a better way to express this without it sounding like a redundancy would be:

He has been taking frequent furloughs since his father's death four months ago.