Azim Ahmed said British Muslims should not be seen as "the solution to extremism and terrorism alone".
But former UK government counter-terrorism minister Kim Howells said Muslim communities should do more to condemn attacks by Islamic terrorists.
The pair were speaking to Sunday Politics Wales.
Mr Ahmed told the programme that politicians "should really start talking differently about British Muslims".
"Not as a suspect community, not as someone who has to start taking care of extremism when no-one else has," he said.
"But really just part of the British public, part of the wider British fabric of society, and not simply look at British Muslims as somehow the solution to extremism and terrorism alone
However, Mr Howells said that there had not been "a huge response" from Muslim communities.
"I haven't seen thousands of British Muslims out on the streets demonstrating against these murders, demonstrating solidarity with people who've been murdered," he said.
"I think communities need to wake up. We've all got a responsibility to try to understand this phenomenon of terrorism and to defeat it. But that includes Muslim societies in this country."
Conservative AM Altaf Hussain said he thought the Muslim community in Wales should be "more open".
A lack of integration, he said, was part of the problem with young people turning to extremist ideology.
"We have created, in the past, multiculturalism which has run its course. We have multiple cultures running parallel not getting integrated," he said.
"That is a problem and we have left it too long... integration is important, we have to be more open."
Expecting Muslim communities to tackle extremism could be counter-productive, the assistant secretary general of the Muslim Council of Wales has claimed.