Speaking in Brussels, he said Slovakia will endeavour to play the part of an "honest broker."
He said that the mandatory quotas proposed by the European Commission as part of reform of the EU's asylum system are only "one element of a very wide process and reform."
The quotas themselves are not expected to feature prominently on the agenda of the Slovakian presidency.
The minister, who has been in his current role since 2012, was speaking at a policy briefing hosted by the European Policy Centre to mark the incoming Slovakian EU Council presidency.
On 1 July - for the first time since its accession to the EU in 2004 - Slovakia will assume the rotating presidency, currently held by the Netherlands.
The EPC event on Wednesday was told it does so at a "difficult" period in European integration, one characterised by multiple crises and a lack of trust between member states.
The Slovak EU Council presidency, said Lajcak, who is also Slovakia's deputy Prime Minister, will seek to play the role of an honest broker, aiming to find consensus and achieve agreements in the Council and also with the European Parliament.
The briefing heard that besides implementing the priorities in the EU presidency trio programme, Slovakia will have to deal with "many" internal and external challenges related to the migration and refugee crisis.
These, it was said, include a revision of the Dublin regulations, the fight against terrorism following the Brussels and Paris attacks, the annual EU budget for 2017, as well as the mid-term review of the Union's multi-annual financial framework 2014-2020.
Depending on the outcome of the UK referendum on 23 June, the Slovak presidency might also have to play a key role in starting the process of a potential exit of Britain from the EU.
Speaking elsewhere, Slovakia's Prime Minister Robert Fico has said that promoting sustainable migrant and asylum policies will be "central" to its term at the helm of the EU.
Slovakia has opposed a compulsory EU plan to redistribute refugees across member states but, speaking after a meeting with European Commission president Jean-Claude Juncker on Wednesday, Fico told reporters he wants "to reach a compromise" whenever possible.