Yes, and in fact several actions were taken, though in practice these tactics largely backfired. I'll summarize Robert Feenstra's 1984 and 1988 papers on the subject below, with a couple additional comments:

The initial attempt to suppress Japanese import competition came in the form of a petition filed by the UAW to the US International Trade Commission for import relief in June 1980; Ford filed a similar petition in August. Essentially, the function of these petitions is to get the ITC to put temporary restrictions on imports to give domestic producers "time to adjust," but one of the conditions for this sort of protection to be given is that the impact of import competition has to be the largest factor causing injury to the domestic industry--and the ITC judged that the contemporary recession in the US was a greater cause of damage to the US auto industry, hence this petition was rebuffed.

In response to the failure of these petitions, legislation was introduced in the Senate in early 1981 which would have imposed an import quota on Japanese autos of 1.6 million units for three years; to get out in front of this, the Japanese agreed to a compromise: a "voluntary export restraint" under which they agreed to a somewhat higher limit of 1,832,500 (including utility vehicles and exports to Puerto Rico) in the first year, with some room for growth in the second year and a review in the third year. At the same time, Japanese compact trucks were subjected to a (fairly hefty) 25% tariff starting in August 1980. Technically, this tariff on trucks had existed since the early 1960s, as a result of an older trade dispute with Germany, but the Japanese had previously circumvented it by selling the trucks without having completed the final assembly, because truck parts were only taxed at 4%, but this loophole was closed.

The result of this policy was twofold: first, you'll notice that the restrictions are on the number of units and say nothing about the value of those cars. So the immediate Japanese corporate response was to increase the unit value of the vehicles they were selling in the United States in order to make the most of the limited quantity they could sell: while some of the observed price increase is simply due to more constrained supply, the Japanese were able to raise the price still further by increasing the quality of the vehicles, adding features such as air conditioning, power steering, increased horsepower, etc. This policy of quality upgrading also eventually led to the creation of the Japanese luxury marques Lexus, Acura, and Infiniti (which until quite recently were not available within Japan). By contrast, American automakers faced fewer incentives to improve their offerings--once the Japanese were capped out, the remainder of the market would be theirs to do with as they pleased. Unfortunately, this probably contributed to the common perception in the 1990s that American cars were no longer well-made, and that you'd be better off buying a Japanese brand even if you had to pay a premium to do so.

The second impact is that Japanese automakers began to look for ways to circumvent the VER: if the Japanese simply manufactured cars inside the United States, the restraint wouldn't apply. The fastest way to get started doing this was to start partnerships with American automakers. This led to a spate of rebadged Japanese vehicles sold under American marques (e.g. the Chevy II Nova, which was basically just a slightly modified Corolla, and the entire Geo lineup). The logical endpoint of this trend is where we're now at: Japanese firms simply have US subsidiaries which manufacture most vehicles for the US market domestically, largely in right-to-work states. This is probably not the outcome that UAW envisioned when they filed that initial petition.

Feenstra, Robert C. "Voluntary export restraint in US autos, 1980-81: quality, employment, and welfare effects." The structure and evolution of recent US trade policy. University of Chicago Press, 1984. 35-66.

Feenstra, Robert C. "Quality change under trade restraints in Japanese autos." The Quarterly Journal of Economics 103.1 (1988): 131-146.