Can I use my old trumpet books for Saxophone?

It depends what kind of book it is. I would say probably not though. If it's a technique/method book, it will contain a lot of flexibility and technique exercises and etudes catered to trumpets. For example, a lot of trumpet books will contain flexibility exercise based on the different "partials" of trumpet playing- for example, holding one fingering and going up and down. These will be completely useless to do on saxophone. If it's a book of "songs" just to play, then it might be fine. Do note, though, that alto saxophone is in a different key than the trumpet. So they won't sound the same, but they'll still form the song. I would recommend getting saxophone books. If you're looking for a method book, the Rubank's books are pretty good.

The tuning is completely different. Trumpets are tuned in Bb, while saxophones are in Eb. You might be able to recognise the melody, but there will be lots of weird sharps and flats thrown in that will sound awful unless you transpose it into your key. Also, the excercizes in beginner trumpet books are for trumpets, like finger and lip excercizes and such, so by using them, even if they are out of tune, you won't benefit on sax. Sax books are best because they will help with getting your fingers and lips adjusted to the saxophone. Also, they'll be in the right key, so you'll be able to tell if you're playing it right.

Trumpet is keyed in Bb. Playing a written C note note will sound Bb. Alto sax is keyed in Eb. Playing a written C will sound Eb.

No, get a sax book! Virtually everything is different, they're not even in the same instrument family.