Recently many mathematicians are working on Genocchi polynomials and Genocchi numbers. We define a new type of twisted <i >q</i>-Genocchi numbers and polynomials with weight <svg style="vertical-align:-0.1254pt;width:8.9375px;" id="M1" height="7.1750002" version="1.1" viewBox="0 0 8.9375 7.1750002" width="8.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.175)"> <g transform="translate(72,-66.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg> and weak weight <svg style="vertical-align:-2.29482pt;width:8.8500004px;" id="M2" height="13.425" version="1.1" viewBox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan> </text> </g> </g> </svg> and give some interesting relations of the twisted <i >q</i>-Genocchi numbers and polynomials with weight <svg style="vertical-align:-0.1254pt;width:8.9375px;" id="M3" height="7.1750002" version="1.1" viewBox="0 0 8.9375 7.1750002" width="8.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.175)"> <g transform="translate(72,-66.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg> and weak weight <svg style="vertical-align:-2.29482pt;width:8.8500004px;" id="M4" height="13.425" version="1.1" viewBox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan> </text> </g> </g> </svg>. Finally, we find relations between twisted <i >q</i>-Genocchi zeta function and twisted Hurwitz <i >q</i>-Genocchi zeta function.