The production of J/$\psi$ is measured at midrapidity ($|y|<0.9$) in proton\textendash proton collisions at $\sqrt{s}$ = 5.02 and 13 TeV, through the dielectron decay channel, using the ALICE detector at the Large Hadron Collider. The data sets used for the analyses correspond to integrated luminosities of $\mathcal{L}_{\rm int}$ = 19.4 $\pm$ 0.4 nb$^{-1}$ and $\mathcal{L}_{\rm int}$ = 32.2 $\pm$ 0.5 nb$^{-1}$ at $\sqrt{s}$ = 5.02 and 13 TeV, respectively. The fraction of non-prompt J/$\psi$ mesons, i.e.\ those originating from the decay of beauty hadrons, is measured down to a transverse momentum $p_{T}$ = 2 GeV/$c$ (1 GeV/$c$) at $\sqrt{s}$ = 5.02 TeV (13 TeV). The $p_{T}$ and rapidity ($y$) differential cross sections, as well as the corresponding values integrated over $p_{T}$ and $y$, are carried out separately for prompt and non-prompt J/$\psi$ mesons. The results are compared with measurements from other experiments and theoretical calculations based on quantum chromodynamics (QCD). The shape of the $p_{T}$ and $y$ distributions of beauty quarks predicted by state-of-the-art perturbative QCD models are used to extrapolate the \bbbar pair cross section at midrapidity and in the total phase space. The total $b\overline{b}$ cross sections are found to be $\sigma_{\rm b \overline{\rm b}} = 541 \pm 45 (\rm stat.) \pm 69 (\rm syst.)_{-12}^{+10} (\rm extr.)~{\rm \mu b}$ and $\sigma_{\rm b \overline{\rm b}}~=~218 \pm 37 (\rm stat.) \pm 32 (\rm syst.)_{-9.1}^{+8.2} (\rm extr.)~{\rm \mu b}$ at \s = 13 and 5.02 TeV, respectively. The value obtained from the combination of ALICE and LHCb measurements in pp collisions at \s = 13 TeV is also provided.