No, the governments of Northern Ireland and the Republic of Ireland cannot agree on unification. Regardless, many people on both sides of the border would support it, and many in the north would violently oppose it. The reality of trying to implement such a thing is, for now, prohibitively complex, dangerous, and expensive.
Northern Irish politics is dominated by the question of whether the province should be part of the United Kingdom or a united Ireland. Northern Ireland's two largest political parties who are/were sharing power, the Democratic Unionist Party, and Sinn Fein, represent the country's [British] unionist and [Irish] nationalist hardliners respectively. Public opinion of which way the province should go is split almost down the middle. There is strong correlation between a protestant background and unionism, just as between a catholic background and nationalism.
Peace in Northern Ireland occured in 1998 following a period of bloody warfare between various paramilitary terror groups and British security forces known as 'The Troubles'. This, and the threat of a return to violence, overshadows all other political concerns locally, and is deeply tied to sectarian identity politics.
South of the border in the Republic of Ireland, it seems the government is less keen than the people on the question of unification. The two largest parties in the Republic, historic rivals Fine Gael and Fianna Fail, are working together recently, largely it seems to spite Sinn Fein. It's important to note that while Sinn Fein is a major power in the north (29.4% of the vote 2017), they are a minor power in the south (13.8% of the vote 2016).
Unification has not been a key issue for most voters in the Republic for a long time. Nonetheless, one poll conducted May 2019 of 3000 Irish voters found that 65% supported unification. To contrast this, in Northern Ireland, unionist parties shared 55% of the vote in the 2017 general election.
Two recent developments have changed things. The apparent inevitability of a no-deal Brexit has led Sinn Fein's leadership in the north to demand a border poll (precursor vote for unification) if no-deal Brexit happens. Sinn Fein has also decided that it will now accept being the minor partner in a coalition government in the Republic.
Leo Varadkar, Ireland's Taoiseach (prime minister), has responded to Sinn Fein by saying Fine Gael will not enter a coalition with them, and also that now would be a bad time for a border poll. Varadkar's objection is that this would require the creation of a new state with a new constitution to take into account the cultural and legal differences with Northern Ireland, and that such an enormous task cannot be rushed.
Add to this the fact that Northern Ireland is one of the most deprived parts of the UK, receives a disproportionate subsidy from Westminster, and has a police-state-like security apparatus required to keep a lid on the north's... unique 'culture' and security risks. Basically, it'd cost the Republic a lot to acquire the north, and in return they'd inherit considerable social and security problems.
In 1994 the Ulster Defence Association released a plan for the "repartition" of Ulster. The UDA, an active loyalist terror group, said that in the event the British Army leaves Northern Ireland, they would withdraw from majority catholic border areas to majority protestant strongholds, and then create a wholly protestant state. This would involve the ethnic cleansing of any catholics who remained.
In 1969 as violence escalated in the north, there was an emergency meeting in the south. The Irish government asked their army if, in a doomsday event where there was a complete breakdown of law and order in the north, they could invade and secure majority catholic border towns Derry and Newry. The army told them no, this was not possible. It seems therefore unlikely that present day security forces from the Republic of Ireland would be able to prevent or suppress the inevitable loyalist insurgency which would result from the unification of Ireland.
Furthermore, the Republic gave up its claim to Northern Ireland as part of the Good Friday Agreement in 1998, upon which peace in Northern Ireland is based. The Good Friday Agreement also affirms that Britain and Ireland must accept the will of the Northern Irish people, whether they wish to remain in the UK or unify with the Republic of Ireland. It is incorrect to suggest that the UK would not permit this, as this is one of the core conditions of peace which they are legally bound to.
There's other political differences at work which make the prospect of unification difficult. The Republic of Ireland has never had a left-wing government, both Fine Gael and Fianna Fail are generally understood to be right-wing nationalists who are more similar than different.
The reality in Northern Ireland is quite different. The public service is a major employer, and both unionists and nationalists are effectively in favour of tax and spend (without the tax; this is Westminster's subsidy).
The situation is complex and changing. The most recent survey of attitudes in Northern Ireland found that half of those surveyed identified as neither nationalist nor unionist, and that this is part of a historic trend away from strong identification with one side or the other.
While unification is unlikely soon, in future it seems possible, some would argue even probable, especially if the protestant community in the north shrinks.
