[Basically I want to know the names of all the teachers of X college so that I can choose what teacher would be perfect for me]
What teacher lectures are you providing?
Whose teacher lectures are you providing?
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Neither one of your sentences is quite correct. It is not common, and sounds odd, to refer to lectures as “teacher lectures.”
You could say, “Whose lectures are you providing?” This is correct because “Whose” already indicates a possessive.
If you don’t use “whose,” you would have to indicate the possessive. You could say “What (or Which) teacher’s lectures are you providing?” Note that I have added the possessive ‘s to “teacher.”