Emission lines indicative of active accretion have been seen on a handful of low-mass companions (M<30MJup) to stars. Line variability is ubiquitous on stellar accretors but has never been characterized in detail on low-mass companions and can give insights on the accretion mechanism at play We aim to characterize the short- to long-term HI Paschen {beta} emission line variability of two 10 to 30MJup companions on wide orbits - GQ Lup b and GSC 06214-00210 b - to clarify the accretion mechanisms at play. We used J-band observations at R=1800-2360, obtained with VLT/SINFONI in 2017, to record time series investigating hours-to-week variability of the HI Paschen {beta} emission line (1.282um). Contrary to H{alpha} it should be less affected by chromospheric activity contamination. The photospheric emission is analyzed at each epoch and removed with the FORMOSA forward modeling tool, using new grids of ATMO models exploring different C/O and [M/H] values. The time series of line profiles and intensities are compared to those of more massive accretors and to predictions from the latest magnetospheric accretion and shock models. To complement, we also re-investigate archival spectroscopic observations at near-infrared wavelengths of each target to increase the time span up to a decade and build a more comprehensive understanding of the variability processes at play. For GQ Lup b, we find line variability on months and decade timescales, while it is within noise level at shorter timescales. For GSC 06214-00210 b, we find line variability on timescales of tens of minutes up to a decade. The line profiles of GSC 06214-00210 b are partly resolved in at least one epoch. Both objects show HI Paschen {beta} flux variability that is moderate (<50%) at timescales below their rotation period, and more important at longer timescales (up to 1000% on decade timescales), a behavior similar to that of classical T Tauri stars. The line profiles of GQ Lup b are blue-shifted and can only be reproduced by magnetospheric accretion models, while GSC 06214-00210 b's are fairly well reproduced by both magnetospheric accretion and shock models, except for one epoch for which the shock model is highly favored. The companions have C/O values broadly consistent with solar values. While magnetospheric accretion is favored for GQ Lup b, higher resolution (R>10000) observations are required to disentangle the two (non-exclusive) emitting mechanisms. The similar variability behavior observed in these low mass companions and in classical T Tauri stars may support similar accretion mechanisms. The high amplitude of variability at months and longer timescales found in both objects could be key to explain the low yield of H{alpha} imaging campaigns.