Aims. We present an analysis of the relation between the star formation rate (SFR) surface density (Σ SFR ) and mass surface density of molecular gas (Σ H 2 ), commonly referred to as the Kennicutt-Schmidt (K-S) relation, on its intrinsic spatial scale, i.e. the size of giant molecular clouds (∼10−150 pc), in the central, high-density regions of four nearby low-luminosity active galactic nuclei (AGN). These are AGN extracted from the NUclei of GAlaxies (NUGA) survey. This study investigates the correlations and slopes of the K-S relation, as a function of spatial resolution and of the different 12 CO emission lines used to trace Σ H 2 , and tests its validity in the high-density central regions of spiral galaxies. Methods. We used interferometric IRAM 12 CO(1−0) and 12 CO(2−1) and SMA 12 CO(3−2) emission line maps to derive Σ H 2 and HST-Hα images to estimate Σ SFR . Results. Each galaxy is characterized by a distinct molecular SF relation on spatial scales between 20 to 200 pc. The K-S relations can be sublinear, but also superlinear, with slopes ranging from ∼0.5 to ∼1.3; slopes are generally superlinear on spatial scales >100 pc and sublinear on smaller scales. Depletion times range from ∼1 and 2 Gyr, which is compatible with results for nearby normal galaxies. These findings are valid independently of which transition -12 CO(1−0), 12 CO(2−1), or 12 CO(3−2) -is used to derive Σ H 2 . Because of either star-formation feedback, the lifetime of clouds, turbulent cascade, or magnetic fields, the K-S relation might be expected to degrade on small spatial scales (<100 pc). However, we find no clear evidence of this, even on scales as small as ∼20 pc, and this might be because of the higher density of GMCs in galaxy centers that have to resist higher shear forces. The proportionality between Σ H 2 and Σ SFR found between 10 and 100 M pc −2 is valid even at high densities, ∼10 3 M pc −2 . However, by adopting a common CO-to-H 2 conversion factor (α CO ), the central regions of the NUGA galaxies have higher Σ SFR for a given gas column than those expected from the models, with a behavior that lies between the mergers or high-redshift starburst systems and the more quiescent star-forming galaxies, assuming that the first ones require a lower value of α CO .