Is there any way to use c++ in chrome extension?

Yes, like with other browsers, you can write a NPAPI plugin that will comunicate with your extension.

See a generic description at wikipedia

NPAPI was deprecated from Chrome/Opera for security concerns.

You can still use NPAPI if you are targeting firefox, but if Firefox follows the trend it might disable it at some point as well.

The better options you have today are

Use NaCl if you are targeting only Chrome

You can use instead NativeClient (A.K.A. NaCL, A.K.A. PPAPI - Pepper Plugin API)

All code ran inside NaCl is sandboxed so it's as secure as a javascript extensions.

Use asm.js if you targeting all Platforms

Using Emscripten you can compile C/C++ code into highly optimized javascript format called asm.js. It is designed to have near native performance and right now has decent support from grade-A browsers except Safari. Though performance is better in Firefox than Chrome at the moment.

Major browser vendors are also working on a new standard called WebAssembly that is partially inspired by asm.js, but not ready for prime usage as of this article

Update 2018-05-14

Native Client is being deprecated and asm.js is being replaced by WebAssembly which is becoming the new standard to compile C++ code for the web.