Error when writing section in an external file

You have to prevent expansion while you write to the file, you can do so by prefixing section with string or noexpand (the difference is small, but with string you'll get no space after section in the output, while with noexpand you'll get section {Test}):

documentclass{article}

makeatletter

newwrite fich@sol
defnameFile{example.sol}
AtBeginDocument{ immediateopenout fich@sol nameFile}
AtEndDocument{ immediatecloseout fich@sol}


begin{document}

Test
immediatewritefich@sol {stringsection{Test}}

makeatother

end{document}