Ratios of cross sections, $\ensuremath{\sigma}(Z+c\text{ }\text{ }\text{jets})/\ensuremath{\sigma}(Z+\text{jets})$, $\ensuremath{\sigma}(Z+b\text{ }\text{ }\text{jets})/\ensuremath{\sigma}(Z+\text{jets})$, and $\ensuremath{\sigma}(Z+c\text{ }\text{ }\text{jets})/\phantom{\rule{0ex}{0ex}}\ensuremath{\sigma}(Z+b\text{ }\text{ }\text{jets})$ in the associated production of a $Z$ boson with at least one charm or bottom quark jet are measured in proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$. The data sample, collected by the CMS experiment at the CERN LHC, corresponds to an integrated luminosity of $35.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, with a fiducial volume of ${p}_{\mathrm{T}}>30\text{ }\text{ }\mathrm{GeV}$ and $|\ensuremath{\eta}|<2.4$ for the jets, where ${p}_{\mathrm{T}}$ and $\ensuremath{\eta}$ represent transverse momentum and pseudorapidity, respectively. The $Z$ boson candidates come from leptonic decays into electrons or muons with ${p}_{\mathrm{T}}>25\text{ }\text{ }\mathrm{GeV}$ and $|\ensuremath{\eta}|<2.4$, and the dilepton mass satisfies $71<{m}_{Z}<111\text{ }\text{ }\mathrm{GeV}$. The measured values are $\ensuremath{\sigma}(Z+c\text{ }\text{ }\text{jets})/\ensuremath{\sigma}(Z+\text{jets})=0.102\ifmmode\pm\else\textpm\fi{}0.002\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\sigma}(Z+b\text{ }\text{ }\text{jets})/\ensuremath{\sigma}(Z+\text{jets})=\phantom{\rule{0ex}{0ex}}0.0633\ifmmode\pm\else\textpm\fi{}0.0004\ifmmode\pm\else\textpm\fi{}0.0015$, and $\ensuremath{\sigma}(Z+c\text{ }\text{ }\text{jets})/\ensuremath{\sigma}(Z+b\text{ }\text{ }\text{jets})=1.62\ifmmode\pm\else\textpm\fi{}0.03\ifmmode\pm\else\textpm\fi{}0.15$. Results on the inclusive and differential cross section ratios as functions of jet and $Z$ boson transverse momentum are compared with predictions from leading and next-to-leading order perturbative quantum chromodynamics calculations. These are the first measurements of the cross section ratios at 13 TeV.