This article deals with the general implementation of 4-component spinor relativistic extended coupled cluster (ECC) method to calculate first order property of atoms and molecules in their open-shell ground state configuration. The implemented relativistic ECC is employed to calculate hyperfine structure (HFS) constant of alkali metals (Li, Na, K, Rb and Cs), singly charged alkaline earth metal atoms (Be+, Mg+, Ca+ and Sr+) and molecules (BeH, MgF and CaH). We have compared our ECC results with the calculations based on restricted active space configuration interaction (RAS-CI) method. Our results are in better agreement with the available experimental values than those of the RAS-CI values.