β-YbAlB4 has become one of the most studied heavy fermion systems since its discovery due to its remarkable physical properties. This system is the first reported Yb-based heavy-fermion superconductor (HFS) for which the low-T superconducting state emerges from a non-fermi-liquid (NFL) normal state associated with quantum criticality Nakatsuji et al 2008 Nature 4 603. Additionally, it presents a striking and unprecedented electron spin resonance (ESR) signal which behaves as a conduction electron spin resonance (CESR) at high temperatures and acquires features of the Yb(3+) local moment ESR at low temperatures. The latter, also named Kondo quasiparticles spin resonance (KQSR), has been defined as a 4f-ce strongly coupled ESR mode that behaves as a local probe of the Kondo quasiparticles in a quantum critical regime, Holanda et al 2011 Phys. Rev. Lett. 107 026402. Interestingly, β-YbAlB4 possesses a previously known structural variant, namely the α-YbAlB4, phase which is a paramagnetic Fermi liquid (FL) at low temperatures Macaluso et al 2007 Chem. Mater. 19 1918. However, it has been recently suggested that the α-YbAlB4 phase may be tuned to NFL behavior and/or magnetic ordering as the compound is doped with Fe. Here we report ESR studies on the α-Yb1-xFexAlB4 (0 ⩽ x ⩽ 0.50) series as well as on the reference compound α-LuAlB4. For all measured samples, the observed ESR signal behaves as a CESR in the entire temperature range (10 K ≲ T ≲ 300 K) in clear contrast with what has been observed for β-YbAlB4. This striking result indicates that the proximity to a quantum critical point is crucial to the occurrence of a KQSR signal.