In a word, no. However, oats do contain a chemical known as avenin which is a compound found in wheat that can cause problems for avenin sensitive celiac patients.
What seems to be a more common issue, however, is cross contamination of oats with wheat, barley, or rye when harvesting, storing, and processing oats because of the similarity of these grains. This was made apparent to me when speaking with the staff of Arrowhead Mills, who take great precautions to insure their gluten free line is not contaminated, and was told that they cannot guarantee their oat products are gluten free because of shared fields and equipment.
If you are a celiac patient or other who would like to continue to have oats in your diet, try Bob’s Red Mill Gluten Free Oats. I have been eating these since my diagnosis without any issue. Bob’s Red Mill sources these oats from farmers who grow only oats, and then test the final product using the R5 Elisa test to insure the oats do not contain gluten.
If you would like to read more about gluten, see this post: What is Gluten?
Diversity in Oat Potential Immunogenicity: Basis for the Selection of Oat Varieties with No Toxicity in Coeliac Disease
Peroral Small Bowel Mucosal Biopsy
Can Oats Be Taken in a Gluten Free Diet? A Systematic Review
Gluten Contamination of Commercial Oat Products in the United States