Unfortunately, there is no option or setting available to show/hide the trend lines. You cannot even control them using calculated fields.

As Hari mentioned, Tableau does not have a built-in on/off switch for trend lines that you can show on a viz, but there is a workaround you might want to consider.

You could do a calculated field for the regression lines, and then create a dual axis chart to overlay the regression line onto your primary chart. A parameter may be used to show/not show the regression line.

I don't have time to do this today, but I  may try it over the weekend. In the meanwhile, here's a link to get you started with plotting regression in Tableau:

https://www.thedataschool.co.uk/emily-dowling/calculate-linear-regression-line-tableau/

Do you have data that you can post?

-- Glenn