Background: The purposes of the study were to: analyze, by objective (accelerometry) and subjective (International Physical Activity Questionnaire, IPAQ) methodologies, the physical activity (PA) and sedentary behavior (SB) in healthy adults (HEALTHY, n = 30) and individuals with primary hypertension (HTN) and overweight/obesity (n = 218); assess the effects of an aerobic exercise intervention on physical activity (PA), sedentary behavior (SB), and sleep quality in the HTN group; and evaluate the relationship between objectively measured and subjectively reported PA and SB. Methods: The measurements were performed before a 16-week exercise intervention period in both HEALTHY and HTN groups and after the intervention period only in the HTN group, randomized to attention control or exercise training (ExT) subgroups. Results: The HEALTHY group showed more moderate-to-vigorous PA (p < 0.05) and better sleep quality (p < 0.05) than the HTN group, but no difference in SB. After the intervention, HTN participants’ PA and SB, objectively measured by accelerometry, were unchanged, but increased PA and decreased SB (p < 0.05) were observed through IPAQ in ExT. The intervention was effective in improving sleep quality in HTN participants. Conclusions: The differences in moderate-to-vigorous PA and SB may be useful in defining the health profile of a population. The supervised aerobic exercise program was effective in increasing PA, reducing SB, and improving sleep quality in overweight/obese adults with HTN. Accelerometer-measured and self-reported data were not comparable, but complementary.