No, we don't use 'lavish' as a synonym for 'generous' in this way. It isn't an attributive adjective, and it doesn't normally describe a personal characteristic of someone.
'Lavish' tends to describe what people might offer, rather than the people themselves - lavish gifts, a lavish meal, lavish praise, lavish lifestyle. Here it means 'extensive' ,'abundant', 'elaborate', 'extravagant' and so on.
The only time we use the word to refer to a person is in a phrase such as 'He was lavish with his hospitality'. This is a predicative rather than an attributive use of the adjective.
It can also be a verb e.g. to lavish affection on someone. Also note the pronunciation - it's pronounced 'lav - ish' rather than 'lave - ish'.