Can I ask a prof (alumni of my college) to recommend me to another prof for a TA position? I was looking for a TA position and mailed prof A, who is an alumni of my college, for a TA. He replied saying that he would not be taking any courses this year so will not require a TA. Can I ask him to recommend me to any other prof, who will need a TA?
Can you do it? Most certainly. Should you do it/is it likely to succeed? That depends on a couple of factors:


Do you know prof A? Does he/she know you? A random email from somebody along the lines of 'Hey, you graduated from my college, please give me a TA position.' will almost always be met with a negative response. 
Have you done some kind of work for prof A in the past? If he/she supervised you at your college for some time, yes, you can ask for a recommendation. If not, again, your cchances of a positive response are slim


Those two points may sound similar, but there is a difference with respect to why you might get a negative response: 
In the first case, it may be the wording of the email, it may be that there is no TA position at the moment. It is mainly to be seen as a general inquiry for a TA position, that you know that professor is only a slight bonus.

In the second case it is a bit different. You will ONLY get a positive response if that professor has supervised you in the past and is satisfied with the work you have done. After all, a letter of recommendation is basically 'This student is good, I have seen it first hand, if you hire him he will do good work for you.' If prof A doesn't know you, he/she will not recommend you because that is putting his/her name at risk.

With these points in mind and the questions I have linked as a comment to your question, you should be able to figure out if that second email is likely to get a positive response.