We use the Cosmic Assembly Deep Near-infrared Extragalactic Legacy Survey (CANDELS) data to study the relationship between quenching and the stellar mass surface density within the central radius of 1 kpc ($Sigma_1$) of low-mass galaxies (stellar mass $M_* lesssim 10^{9.5} M_odot$) at $0.5 leq z < 1.5$. Our sample is mass complete down to $sim 10^9 M_odot$ at $0.5 leq z < 1.0$. We compare the mean $Sigma_1$ of star-forming galaxies (SFGs) and quenched galaxies (QGs) at the same redshift and $M_*$. We find that low-mass QGs have higher $Sigma_1$ than low-mass SFGs, similar to galaxies above $10^{10} M_odot$. The difference of $Sigma_1$ between QGs and SFGs increases slightly with $M_*$ at $M_* lesssim 10^{10} M_odot$ and decreases with $M_*$ at $M_* gtrsim 10^{10} M_odot$. The turnover mass is consistent with the mass where quenching mechanisms transition from internal to environmental quenching. At $0.5 leq z < 1.0$, we find that the $Sigma_1$ of galaxies increases by about 0.25 dex in the green valley (i.e., the transitioning region from star forming to fully quenched), regardless of their $M_*$. Using the observed specific star formation rate (sSFR) gradient in the literature as a constraint, we estimate that the quenching timescale (i.e., time spent in the transition) of low-mass galaxies is a few ($sim4$) Gyrs at $0.5 leq z < 1.0$. The mechanisms responsible for quenching need to gradually quench star formation in an outside-in way, i.e., preferentially ceasing star formation in outskirts of galaxies while maintaining their central star formation to increase $Sigma_1$. An interesting and intriguing result is the similarity of the growth of $Sigma_1$ in the green valley between low-mass and massive galaxies, which suggests that the role of internal processes in quenching low-mass galaxies is a question worthy of further investigation.
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