No. Denmark never had the resources or manpower, not to speak of the economy, to compete with the great colonial powers at that time, and for the same reasons no intentions of colonising in that direction. They simply had enough of a challenge exploiting the vast regions of Greenland, Iceland and the Faroe Islands which they possessed and could exploit for resources.

Their far reaching colonial economic efforts during the early colonial period was mainly invested in getting a share of the triangle trade which resulted in the possessions at the Gold Coast of Africa and a group of islands in the West Indies as the main investment subjects. They also eventually gained trade dominions in India and the East Indies for the same reasons, and like the West Indian dominions they were mainly strictly for trade purposes. That is Danish military and administrative efforts, but no real colonisation of settlers, as Denmark simply did not have the population for those kind of endeavours at that time.

But they did send out the explorer Jens Munk to discover the North West Passage to East India in 1619. They did not succeed in their mission, but did succeed in being the first to penetrate as far as the entrance to Hudson Bay before being forced to turn back.