No, probably not.
The word "order" in English has many different meanings. The English Oxford Dictionary lists nine of them.
"Order" in "Order 66" uses the Oxford Dictionary's second meaning: "command, instruction, request." Before The Clone Wars animated series introduced the bio-chip story arc, Order 66 was a part of the Senate-approved document "Contingency Orders for the Grand Army of the Republic: Order Initiation, Orders 1 Through 150." Some of video games and books featuring clone troopers provided insight into the document and quoted some other orders that the soldiers had to know. Unfortunately, this is Legends now.
"Order" in "First Order" uses the Oxford Dictionary's fourth meaning: "A society of monks, nuns, or friars living under the same religious, moral, and social regulations and discipline." Consider the sub-meaning 4.1 in the light of the fact, that Snoke was overseer of both Knights of Ren and First Order.
It doesn't have to mean anything, but the Polish translation translated "First Order" into "Najwyższy Porządek.” In this context, the word chosen for "First" directly translates to "Highest". However, I'm not sure if official translators were provided with some undisclosed information in order (sic!) to provide a more accurate translation, or that was just the translator's gut feeling.