The purpose of this paper is to assess the effect of glucagon-like peptide-1 receptor agonists (GLP-1RAs) on stroke or heart disease in patients having chronic respiratory disease and diabetes (CD) with underlying diseases related to COVID-19. From 1998 to 2019, we adjusted competing risk by assessing the effect of GLP-1RAs on stroke or heart disease in a CD cohort after propensity matching based on the Taiwan National Health Insurance Research Database. We also used the time-dependent method to examine the results. GLP-1 RA and non-GLP-1 RA user groups included 15,801 patients (53% women and 46% men with a mean age of 52.6 ± 12.8 years). The time between the diagnoses of DM and the initial use of the GLP-1 RA among the stroke subcohort (<2000 days) was shorter than that of the heart disease subcohort (>2000 days) (all p-values < 0.05). The overall risks of stroke, ischemic, and hemorrhagic stroke were significantly lower in GLP-1 RA users than nonusers. The adjusted subhazard ratio (aSHR) was 0.76 [95% CI 0.65-0.90], 0.77 [95% CI 0.64-0.92], and 0.69 [95% CI 0.54-0.88] (p < 0.05 for all). Furthermore, a ≥351-day use had a significantly lower stroke risk than GLP-1 RA nonusers (aSHR 0.35 [95% CI 0.26-0.49]). The time-dependent method revealed the same result, such as lower stroke, and ischemic or hemorrhagic stroke risk. In contrast, the cardiac arrhythmia incidence was higher in GLP-1 RA users with an aSHR of 1.36 [95% CI 1.16-1.59]. However, this risk disappeared after the ≥351-day use with 1.21 (0.98, 1.68) aSHR. Longer GLP-1 RA use was associated with a decreased risk of ischemic or hemorrhagic stroke and the risk of cardiac arrhythmia disappears in a CD cohort. Both a shorter lag time use of the GLP-1 RA and a longer time use of GLP-1 RA were associated with a decreased risk of ischemic or hemorrhagic stroke in the CD cohort. The GLP-1 RA use in the early stage and optimal time use in the CD cohort may avoid the stroke risk.