Despite recent squabbling over the qualifications of the company picked to bring cable television to Alexandria, the City Council voted unanimously last night to enter a formal contract with the firm, Alexandria Cablevision.

The company, expected to make cable TV service available by 1981, has a number of locally prominent persons as its officers or backers, including former Democratic House of Delegates Majority Leader James M. Thomson; fomer City Council member James W. Carroll; Harry S. Flemming who was an aide in the Nixon Administration, and local publisher James Coldsmith.

Last night's vote came after Deputy City Manager Clifford Rusch said he "regretted" saying in a recent memo that city consultants were still unable t0 "make a definitive judgement" that the company "can or cannot meet its meeting.

Based on those documents, two city consultants and the city cable TV administrator told the councl they were satisfied with the company's ability to fund the $6.5 million system and begin operation in 18 months.

The previous council, at its last meeting before leaving office last month, awarded the company a franchise to use a city rights-of-way in laying cable. That award was made only after the city forced the firm to come up with an additional $1 million and found a $655,000 "mathematical error" in its figures.

Since then, the company's qualifications have been hotly debated; former Vice Major Nora Lamborne urged her successors to reconsider the issue.

The contract agreed to last night was needed to bind the company to the promise its has made regarding local programming and service to apartments and private homes, according to City Attorney Cyril D. Calley.