It would have been hokey, contrived, and downright absurd. But for some odd reason a lot of people seem to be enamored with the idea of Wesley being the result of an adulterous tryst between Picard and Beverly. Let's look at just a few of the ramifications of Wesley being Jean-Luc Picard's and Beverly Crusher's love child.
This means characterizing Picard as a man willing to bed his best friend and fellow officer's wife. No way around it. I know P/C is a favorite shipper pairing to a lot of people so take it out of that context. This is the equivalent of Picard sleeping with Deanna Troi after her marriage to Will Riker in Nemesis.
This would also be characterizing Beverly Crusher as a woman willing to betray her husband with his best friend and commanding officer. (Again, think how it would change Deanna's character if you have her bed Jean-Luc Picard while married to Will Riker.) Beverly would also have to be willing to lie to her husband passing the child of an adulterous tryst off as his.
Warmest Wishes,
Whoa Nellie