Please create a Integer field on Quote with API name "Min_of_Age_Updated__c".

And then try below trigger and class.

FYI- I have assumed API of lookup field as Quote__c.

******************************************************************Trigger*******************************************************

trigger DealTrigger on Deal__c (after delete,after update, after insert, after undelete)
{
    if(trigger.isInsert)
    {
        OpportunityHandlers oppHandler =new OpportunityHandlers();
        oppHandler.oppInsert(Trigger.New); 
    }
    else if(trigger.isDelete)
    {
        OpportunityHandlers oppHandler =new OpportunityHandlers();
        oppHandler.oppDelete(Trigger.Old);
    }
    else if(trigger.isUnDelete)
    {
        OpportunityHandlers oppHandler =new OpportunityHandlers();
        oppHandler.oppundelete(Trigger.New);
    }
    else if(trigger.isUpdate)
    {
        OpportunityHandlers oppHandler =new OpportunityHandlers();
        oppHandler.oppUpdate(Trigger.New,Trigger.Oldmap);
    }

}

*******************************************************************Class***************************************************************************************************************************

public class OpportunityHandlers
{
    list<Quote> oppList = new list<Quote>();
    set<id> oppIDs = new set<id>();

    
    public void oppInsert(List<Quotation_Driver__c> oppLists)
    {
        for(Quotation_Driver__c opp : oppLists)
        {
            if(opp.Quotation_Driver__c !=null)
            {
                oppIDs.add(opp.Quotation_Driver__c);
            }
        }
        aggregateResults();
    }
    
    public void oppUpdate(List<Quotation_Driver__c> oppLists,Map<Id,Quotation_Driver__c> oppOldmap)
    {
        for(Quotation_Driver__c opp : oppLists)
        {
            if(opp.Quote__c!=null)
            {
                oppIDs.add(opp.Quote__c);
                if(oppOldmap.get(opp.id).Quote__c!=opp.Quote__c)
                {
                    oppIDs.add(oppOldmap.get(opp.id).Quote__c);
                }
            }
             
        }
        aggregateResults();

    }
    
    public void oppDelete(List<Quotation_Driver__c> oppLists)
    {
        for(Quotation_Driver__c opp : oppLists)
        {
            if(opp.Quote__c!=null)
            {
                oppIDs.add(opp.Quote__c);
            }
        }
        aggregateResults();
    }
    public void oppundelete(List<Quotation_Driver__c> oppLists)
    {
        for(Quotation_Driver__c opp : oppLists)
        {
            if(opp.Quote__c!=null)
            {
                oppIDs.add(opp.Quote__c);
            }
        }
        aggregateResults();
    }
    
    public void aggregateResults()
    {
        AggregateResult[] groupedResults = [SELECT MIN(AgeUpdated__c), Quote__c FROM Quotation_Driver__c where Quote__c IN :oppIDs GROUP BY Quote__c];
        for(AggregateResult ar:groupedResults) 
        {
            Id oppId = (ID)ar.get('Quote__c');
            Integer min = (INTEGER)ar.get('expr0');
            Quote opp1 = new Quote(Id=oppId);
            opp1.Min_of_Age_Updated__c= min;
            oppList.add(opp1);
        }
            if(oppList.size()>0 && oppList!=null)
            {
                update oppList;
            }
    }
}