A (1,k)-overlap-free code, motivated by applications in DNA-based data storage systems and synchronization between communication devices, is a set of words in which no prefix of lengthtof any word is the suffix of any word for every integer t such that 1 ≤t≤k. A (1,n— 1)-overlap-free code of lengthnis said to be non-overlapping. We provide a construction forq-ary (1,k)-overlap-free codes of length 2k, which can be viewed as a generalization of the Zero Block Construction presented by Blackburn, Esfahani, Kreher and Stinson recently over a binary alphabet, and analyze the asymptotic behavior of their sizes. Whenn≥ 2k, an explicit general lower bound and an asymptotic lower bound for the size of an optimalq-ary (1,k)-overlap-free code of lengthnare presented. The exact value of the maximum size ofq-ary (1, 2)-overlap-free codes of lengthnis determined for anyn≥ 4, and a construction forq-ary (1,k)-overlap-free codes of lengthk+ 2 is given.