Mr Quinn does not want a silent majority to determine the outcome of what is a historic opportunity to reshape the primary school landscape.
The Department of Education would not say what level of response they had received to surveys to date, but Mr Quinn is keen to achieve as high a turnout as possible and to have a clear picture of parental preferences.
Parents in five areas have until next Friday to have their say on whether they want to hand over any Catholic-run schools to other patron bodies.
The surveys, being conducted in Arklow, Co Wicklow; Trim, Co Meath; Whitehall, Dublin; Castlebar, Co Mayo; and Tramore, Co Waterford, are open to parents of pre-school and primary-aged children.
The move is designed to reflect the increasing social and religious diversity in Irish society and to give parents an opportunity to reduce the dominance of the Catholic Church in primary education.
The Catholic Church, which controls about 90pc of primary schools, agrees that greater choice is required and is willing to hand over schools in line with parental preferences.
Similar surveys are planned for a further 39 towns and suburbs next month, and the areas have been selected because virtually all of the local primary schools are Catholic-run.
Alternative
The responses of parents will be used to decide what changes, if any, should be made to widen the availability of other types of primary school in these locations.
If the surveys identify demand for alternative patronage in these areas, the Department of Education will explore, with the existing patrons, the transfer of Catholic schools.
Mr Quinn is concerned that a poor response to the survey could cause some confusion as to what parents actually want.
Fr Michael Drumm, of the Catholic Schools Partnership, has painted a scenario of only a relatively small number of motivated parents participating in the survey and expressing a desire for change.
Fr Drumm said of those who did not participate: "We would have to conclude (they) are actually content with the present situation."
But in a message to parents in the five areas last night, Mr Quinn urged all those eligible "to make sure that their voices are heard by taking part in these surveys".
It is impossible to predict how many Catholic schools will be handed over under this process, but Mr Quinn wants to name the first batch by next June.
The online survey is available at: www.education.ie and parents or guardians are advised to have their PPS number to hand for validation purposes.
If parents cannot access the online survey, a paper-based version can also be requested by ringing the freephone helpline number on: 1800 303621.