Depends on the grains. Many of the ancient grains, such as polenta, spelt, millet, and farro, are high in protein. Once difficult to find in modern grocery stores, they are more readily available now and can always be purchased online.
Complex grains, not the highly processed ones like white flour, are slowly digested, help you feel full longer, and can help you lose weight and maintain a stable energy level without the highs and lows caused by highly processed grains.
I have kept food diaries on and off for years and even in times when I was not making any effort to buy the healthiest vegetarian foods, I usually was near or over the daily recommended amount of protein.
Thanks for reading and commenting and good luck with your vegetarian diet.