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Recently it was found that selenium doping can suppress thecharge-density-wave (CDW) order and induce bulk superconductivity in ZrTe$_3$.The observed superconducting dome suggests the existence of a CDW quantumcritical point (QCP) in ZrTe$_{3-x}$Se$_x$ near $x \approx$ 0.04. To elucidateits superconducting state near the CDW QCP, we measure the thermal conductivityof two ZrTe$_{3-x}$Se$_x$ single crystals ($x$ = 0.044 and 0.051) down to 80mK. For both samples, the residual linear term $\kappa_0/T$ at zero field isnegligible, which is a clear evidence for nodeless superconducting gap.Furthermore, the field dependence of $\kappa_0/T$ manifests multigap behavior.These results demonstrate multiple nodeless superconducting gaps inZrTe$_{3-x}$Se$_x$, which indicates conventional superconductivity despite ofthe existence of a CDW QCP.