Yes, protein can have an effect on your bowel movements. Protein-rich foods like fish, meat, nuts, and eggs are generally more difficult to digest than carbohydrates and fats, which can lead to changes in your GI tract that lead to constipation or diarrhea.
Protein can also affect the speed at which food moves through the intestine. High amounts of dietary protein promotes water absorption in the small intestine, stiffening the stool and thus making it harder for it to pass through your digestive system quickly. This can result in increased transit time - or longer bowel movements - as well as a decrease in colonic fermentation leading to decreased gas production and less irritation of the GI tract lining that cause bloating or cramping.
Some people find that a high-protein diet helps with regularity due to its diuretic effects but this isn't true for everyone so if you're noticing any adverse effects after eating protein rich meals it might be best for you speak with a medical professional about adjusting your diet accordingly.