I have owned both red ear sliders and yellow sliders in addition to beardies, but never at the same time. I know that beardies are definitely effected by other animals within view of them, so I'd imagine that turtles are exactly the same way, as I know my Russian Tortoise that I had for 22 years and rehomed to my cousin (he's a sweetie) hated my parrots and I had to disassemble his Tortoise table and move it to a room that was completely off-limits to my birds. He had a huge Tortoise table that I built him that went around the room, was two levels tall, and was open with no lids or top. When he would see one of my parrots, particularly my mom's large Congo African Gray, he would move as fast as he could "run", lol, to a spot under the second level out of the light and out of sight, and often burrow his head under the sand in that area.
So if your turtle just started acting this way the minute you moved the beardie across the room from him, then I'd guess that it's not a coincidence and the beardie is stressing him out, which is bad for his health and can eventually effect his appetite and basking behavior. What I would do is move your new BEARDIE to a different room, not your turtle, because your beardie is new and can handle the stress, where as your turtle has been in that room much longer, he's used to it, it's his home, and plus he's already stressed out, so more stress from moving him to a different room after all the time he's been there could cause him even more issues. After you move the beardie to a different room and out of sight, wait a week or so and make sure your turtle starts acting normally again. However, if after you move your beardie to a different room and out of sight, your turtle continues to act strangely and doesn't go back to his "old self", or god forbid he gets worse and stops eating or starts any other odd behavior, then he needs to go see an experienced reptile vet ASAP because he could be sick and this timing with the new beardie could just be a coincidence. Though I'm pretty sure that it is the arrival of the new beardie in his room that is causing the problem.
