This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring R=Fq+uFq+vFq+uvFq, where u2=v2=0,uv=vu, and Fq is a finite field of cardinality q=pr; p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over Fq are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in Fq and Fq+uFq(u2=0) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature.