Learn French: Le suspect que vous interrogez a un alibi. - The suspect you are interrogating has an alibi.

No, you can never drop the relative pronoun in French. Even though English lets us skip words like "that" or "whom" in the middle of a sentence, French requires que to connect the noun (le suspect) to the action describing it (vous interrogez).

Why is it just vous interrogez instead of a literal translation for "you are interrogating"?

French doesn't have an exact equivalent for the English "to be doing something" tense (the present continuous). The standard present tense vous interrogez covers both "you interrogate" and "you are interrogating".

What is the main verb of this sentence?

The main verb is a (has). The core of the sentence is simply Le suspect a un alibi (The suspect has an alibi). The phrase que vous interrogez is just inserted to give more detail about the suspect.

Are suspect and alibi exactly the same in French and English?

Yes, both are "true cognates"—words that look the same and mean the same thing in both languages. Just remember to use French pronunciation, and note that both words are masculine: le suspect and un alibi.