To augment what others have already said, it really depends on what you mean by "Linux Programming".
E.g., you can do programming using Bash, which is a command-shell script program. The most typical use is to act as the mediator between the human, who enters a command, and the computer, which executes the command, then displays the results of that command.
Additionally, the script can be set up to run certain commands, as if typed by the user, according to certain conditions. E.g., it can run a compiler against source code, and the stop processing if the compiler detects an error condition. Or, if no errors are found, the output of the compiler can be passed to a linker to link various modules together into an application. Then finally, if no errors occur in linking, the result can be installed into a certain directory for public use.
All of this can be done without knowing C or C++, and scripts can be just as complex.
BUT -- --
If you mean "Linux Programming" as in programming modules that will become *part* of Linux, such as device drivers, or security controls, or networking, etc. etc., then yes, you must learn C. I would guesstimate that probably 92% of that is written in C, 3% in C++, and the rest in assembly and machine language.
You are not going to write that kind of Linux Programming without knowing C and C++.