A computer expert was someone who was extremely knowledgeable in the use of computers.
In 2267, when Commodore José I. Mendez disputed Spock's claims to have received a message from Starbase 11, James T. Kirk suggested that the record tapes Mendez cited had been deliberately altered by a computer expert, who could "change record tapes, duplicate voices, say anything, say nothing." (TOS: "The Menagerie, Part I")
As of 2268, Spock held a A-7 computer expert classification, rendering him qualified to comment on the M-5 multitronic unit. (TOS: "The Ultimate Computer")
In 2375, while ruminating on how many Section 31 personnel would have to be involved in a conspiracy to infect Odo with a morphogenic virus, Julian Bashir surmised that at least 73 computer experts, doctors, security officers, admirals and clerks would have to be involved. (DS9: "Extreme Measures")