Is it possible to execute adb commands through my android app?

You can do it with this:

Process process = Runtime.getRuntime().exec("your command");
BufferedReader bufferedReader = new BufferedReader(
new InputStreamReader(process.getInputStream()));

Don't forget to surround it with a try and catch statement.

Edit:

@Phix is right, ProcessBuilder would be better to use.

Normal Android apps have different privileges to processes started via adb, e.g., processes started via adb are allowed to the capture the screen whereas normal apps aren't. So, you can execute commands from your app via Runtime.getRuntime().exec(), but they won't have the same privileges as if you had executed from an adb shell.