I want to refer to the timeframe of Summer 2019 in the Northern hemisphere. However, the writing is intended for a global audience, and when it is Summer in the Northern hemisphere it will be Winter in the southern hemisphere.

Is there terminology I can use that refers to that period of time in both hemispheres?

The common phrase is

northern summer.

While readers in the northern hemisphere may understand this to more narrowly mean summer in the northern parts of whichever continent they live on – as in the title of Sir John Carr's book A Northern Summer; Or, Travels Round the Baltic, Through Denmark, Sweden, Russia, Prussia, and Part of Germany, in the Year 1804 –, readers in the southern hemisphere will think of this in the way you intend, and certainly none of them will think of Africa or Australia.

In the natural sciences, "northern summer" does have exactly the meaning you seek, as can be seen in this quote from Jaffe & Taylor's The Physics of Energy from 2018:

And when the precession index e sin 𝜛 is at a maximum, so that Northern summer coincides with perihelion while eccentricity is at a maximum, the Northern Hemisphere also receives more summer insolation.

There are countless examples for this use in the sciences, but it is maybe more interesting to note that the term is used by Australian institutions when referring to what you might call "global seasons", as in this Australian government website on international aviation (my emphasis):

The timetable summary provides information on airlines approved to operate scheduled services between Australia and its bilateral partners for the Northern Winter period from 28 October 2018 to 30 March 2019.

There are a few simple ways to do that.

1:'the second season of the year', it feels a tad uncomfortable but it works.

2: you can simply say which month or date it is. Regardless of where you are in the world the dates stay the same* (so December is warm in the southern hemisphere while July is cold).

3: 'halfway through the year' would be Summer in the north and winter in the south.

*other than the international date line, but that is just a 1 day difference so it doesn't really make a difference.

edit: user10915156 has a better answer, go upvote his

Many global corporations just say Q1, Q2, Q3. So Q3 2019.

In your comments you state

it's a piece of technical writing that will be read globally

and

I very specifically need to refer to the timeframe of June-July-August.

Given that this is technical writing you really shouldn't be stating this as a season. This would be true even if your target audience is local, but is even more important for a global audience. It isn't specific enough. When you need to be specific, just be specific. Given that you like how "Summer 2019" "rolls off the tongue" might I suggest the following:

• June through August 2019


• June, July, and August 2019


• June-August 2019

There is no reason to try to get fancy by stating this as a season, since seasons aren't going to convey a specific date range very accurately. I would also highly recommend against using "northern summer", as another answer suggests, as this will just confuse many of your readers who don't know what you are talking about. Even for those readers who know what you mean, it still suffers from a lack of specificity, which you have indicated is unwanted in this case.