The polynomials $f_{i,j}$ are introduced by Abe-Harada-Horiguchi-Masuda to produce an explicit presentation by generators and relations of the cohomology rings of regular nilpotent Hessenberg varieties. In this paper we quantize the polynomials $f_{i,j}$ by a method of Fomin-Gelfand-Postnikov. Our main result states that their quantizations $F_{i,j}$ are related to the coordinate rings of regular semisimple Hessenberg varieties. This result yields a connection between the coordinate rings of regular semisimple Hessenberg varieties and the cohomology rings of regular nilpotent Hessenberg varieties. We also provide the quantized recursive formula for $F_{i,j}$.