Background: Ultra-low $Q$-value $beta$-decays are interesting processes to study with potential applications to nuclear $beta$-decay theory and neutrino physics. While a number of potential ultra-low $Q$-value $beta$-decay candidates exist, improved mass measurements are necessary to determine which are energetically allowed. Method: Penning trap mass spectrometry was used to determine the atomic mass of $^{89}$Y and $^{139}$La, from which $beta$-decay $Q$-values for $^{89}$Sr and $^{139}$Ba were obtained to determine if there could be an ultra-low $Q$-value decay branch in the $beta$-decay of $^{89}$Sr $rightarrow$ $^{89}$Y or $^{139}$Ba $rightarrow$ $^{139}$La. Results: The $^{89}$Sr $rightarrow$ $^{89}$Y and $^{139}$Ba $rightarrow$ $^{139}$La $beta$-decay $Q$-values were measured to be $Q_{rm{Sr}}$ = 1502.20(0.35) keV and $Q_{rm{Ba}}$ = 2308.37(68) keV. These were compared to energies of excited states in $^{89}$Y at 1507.4(1) keV, and in $^{139}$La at 2310(19) keV and 2313(1) keV to determine $Q$-values of -5.20(37) keV for the potential ultra-low $beta$-decay branch of $^{89}$Sr and -1.6(19.0) keV and -4.6(1.2) keV for those of $^{139}$Ba. Conclusion: The potential ultra-low $Q$-value decay branch of $^{89}$Sr to the $^{89}$Y (3/2$^-$, 1507.4 keV) state is energetically forbidden and has been ruled out. The potential ultra-low $Q$-value decay branch of $^{139}$Ba to the 2313 keV state in $^{139}$La with unknown J$^{pi}$ has also been ruled out at the 4$sigma$ level, while more precise energy level data is needed for the $^{139}$La (1/2$^+$, 2310 keV) state to determine if an ultra-low $Q$-value $beta$-decay branch to this state is energetically allowed.
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