Yes. This is now possible thanks to the addition of Tasha's Cauldron of Everything. You can select the Fey Touched feat in order to learn a "1st-level spell... from the divination or enchantment school of magic."
Since Hex is an enchantment spell and Hunter's Mark is a divination spell and both are first level, you may pick either one.
This feat also grants +1 to Intelligence, Wisdom, or Charisma as well as learning Misty Step. Both spells may be cast once per day using this feat as well as with spell slots if you have them.
Everything below here is the old answer, which used to be correct but is now obsolete and wrong.
No, there does not appear to be a way for Eldritch Knights to learn cross-class spells that work with their slots.
This answer used to suggest it was allowed based on a strict reading of the grammar used in the Eldritch Knight subclass writeup. This seemed possible due to text which allowed you to use slots to cast spells that you knew. However, PHB errata has since rewritten that text so that it explicitly states that you may use slots to cast wizard spells that you know.
Since these spells are not wizard spells you may not use your spell slots to cast them. You only gain the ability to cast warlock spells with the warlock spellcasting feature and the same thing for ranger spells. Therefore, without multiclassing, these spells are inaccessible to you to cast with spell slots.
It's not quite the same as hex and hunter's mark, but you might look into **Shadow Blade** (*XGtE p. 164*). It's a 2nd-level illusion spell that you can learn at level 8 with your unrestricted wizard spell choice. It creates a powerful blade dealing 2d8 psychic + str/dex on every hit. It only lasts for a minute and if your concentration is broken, you just lost your weapon. However, it has finesse, light, and thrown properties (reform as a bonus action).