Can I make a monster indestructible?
Based on what you describe, the card that comes to mind that better fits your needs is Safe Zone:


  Activate this card by targeting 1 face-up Attack Position monster on the field; that face-up monster cannot be targeted or destroyed by your opponent's card effects, or be destroyed by battle. It cannot attack your opponent directly. When this card leaves the field, destroy that monster. When that monster leaves the field, destroy this card.


emphasis mine

Of course, every card has it's trade-offs, and this one's are that you can't attack directly. Also, if Safe Zone is removed that monster is also removed (however, if you are talking about Divine Dragon Knight Felgrand, he can save himself from this destruction with his own effect). 

Now, it seems that you have to protect your Safe Zone from being destroyed. One card that can do that is Imperial Custom:


  Face-up Continuous Trap Cards cannot be destroyed, except "Imperial Custom". You can control only 1 "Imperial Custom".




Another way you can make a monster card impervious to any other card is with the effect of Number 81: Superdreadnought Rail Cannon Super Dora:


  Once per turn, during either player's turn: You can detach 1 Xyz Material from this card, then target 1 face-up monster on the field; that target is unaffected by card effects, except for its own effects, until the end of this turn.


This will prevent basically any target, effect destruction, change of control, or other effects that do not target. Of course, if your opponent summons more muscle than your card's it can still go down by battle. I suppose that combining Safe Zone with Number 81's effect would really make it hard to touch. 

Other, option you got similar to Number 81 is Divine Dragon Knight Felgrand, which does basically the same plus negating that monster's effect (which could come handy in other scenarios):


  Once per turn, during either player's turn: You can detach 1 Xyz Material from this card, then target 1 face-up monster on the field; this turn, its effects are negated, but it is unaffected by other card effects.