We present the Lyman-α (Lyα) luminosity function (LF) at 2.05 < z < 3.75, estimated from a sample of 67 Lyα-emitter (LAE) candidates in the Javalambre Physics of the Accelerating Universe Astronomical Survey (J-PAS) pathfinder surveys: miniJPAS and J-NEP. These two surveys cover a total effective area of ∼1.14 deg2 with 54 narrow band (NB) filters (FWHM ∼ 145 Å) across the optical range, with typical limiting magnitudes of ∼23. This set of NBs allowed us to probe Lyα emission in a wide and continuous range of redshifts. We developed a method for detecting Lyα emission for the estimation of the Lyα LF using the whole J-PAS filter set. We tested this method by applying it to the miniJPAS and J-NEP data. In order to compute the corrections needed to estimate the Lyα LF and to test the performance of the candidate selection method, we built mock catalogs. These include representative populations of LAEs at 1.9 < z < 4.5 as well as their expected contaminants, namely low-z galaxies and z < 2 quasi-stellar objects (QSOs). We show that our method is able to provide the Lyα LF at the intermediate-bright range of luminosity (43.5 ≲ log10(LLyα/erg s−1) ≲ 44.5) combining both miniJPAS and J-NEP. The photometric information provided by these surveys suggests that our samples are dominated by bright, Lyα-emitting active galactic nuclei (i.e., AGNs). At log10(LLyα/erg s−1) < 44.5, we fit our Lyα LF to a power law with a slope of A = 0.70 ± 0.25. We also fit a Schechter function to our data, obtaining the following: log10(Φ∗/Mpc−3) = −6.30−0.70+0.48, log10(L∗/erg s−1) = 44.85−0.32+0.50, and α = −1.65−0.27+0.29. Overall, our results confirm the presence of an AGN component at the bright end of the Lyα LF. In particular, we find no significant contribution of star-forming LAEs to the Lyα LF at log10(LLyα/erg s−1) > 43.5. This work serves as a proof of concept for the results that can be obtained with the upcoming data releases of the J-PAS survey.