If the Martells were using roads which are open to all (see below), then probably not. However, if the Martells were trying to use a proprietary bridge or toll road or cross someone’s fields without their permission, there the Tyrells would have more of a leg to stand on.
magicbeardpowers asks:
If an individual lord or group of lords wanted to put a proper road through their land, would they have to get the king’s or lord paramount’s permission, like they would a city charter or to build a dam?
You did see attempts by lords to exert authority over roads on their lands – usually by trying to levy tolls – but as time went on, kings successfully asserted the legal principle that main roads were considered “public thoroughfares” and under the protection of the king, and then in the later Middle Ages merged that concept with nuisance law, whereby obstructions, enclosures, or interference with public thoroughfares were considered to be injurious to the commonweal and damage to royal property, which gave the king two separate avenues for exerting authority over roads.
However, note the use of the word “public” there. Whether a road was considered a public thoroughfare could depend on certain factors: did the road lead to a town, port, or market, was the road listed as particularly belonging to the king (Fosse Way, Ermine Street, Watling Street, and the Icknield Way were called the “king’s four highways” in English law), and had it traditionally been open to the public.
But to answer your question, I would guess on principle that if the road was contained within the fiefdom of a lord and hadn’t traditionally been open to the public and it wasn’t a road that the overlord had asserted ownership over, they probably could do it without permission.
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