TL;DR: the +1 is tied to the dagger's damage roll
The +1 bonus damage is applied to damage rolls "made with this magic weapon." The class ability's 1d6 roll is being substituted for the weapon's damage roll.
You can roll a d4 {now 1d6 at lvl 5} in place of the normal damage of your unarmed strike or monk weapon. The 1d6 roll replaces the monk weapon damage roll. (Staff, dagger, short sword, whatever...)
Discussion
Damage Rolls
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage. With a penalty, it is possible to deal 0 damage, but never negative damage. When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage.
An analysis of the rules lends support to the ruling that the "normal damage" of the +1 dagger is the quantity "1d4+1." The way the martial arts text reads, you make a choice of which roll to use in the form of either / or: either sub in the Monk class feature damage roll (per table) or use the weapon's damage roll.
You can roll a d4 {now 1d6 at your lvl 5} in place of the normal damage of your unarmed strike or monk weapon (Fuller text from rules at the bottom)
The +1 bonus looks to be inherent in the weapon. I found nowhere in the rules an indication that the +1 bonus is transferable to another weapon or attack, melee or otherwise, for a simple +X weapon. A notable exception is the Defender Sword that can transfer + to hit/damage to a + to armor class. (That's a legendary item that requires attunement).
(SRD pg 254/DMG p. 213) Weapon, +1, +2, or +3 uncommon (+1), rare (+2), or very rare (+3)
You have a bonus to attack and damage rolls made with this magic weapon. The bonus is determined by the weapon’s rarity.
From the way that reads, if you choose the magic weapon, its damage (1d4+1) is used, and modified by a Dex/Str bonus to arrive at total damage (see above). If you choose the martial art feature, use that damage roll (1d6) which modified by Dex/Str to arrives at total damage.
- For what it's worth, that 1d4+1 averages to the same amount of damage as 1d6, (3.5) but you won't get any 6's or 1's.
From the Martial Arts rules text
At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are short swords and any simple melee weapons that don’t have the two‑handed or heavy property. You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:
• You can roll a d4 {now 1d6 at lvl 5} in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table. {you can choose to sub in the unarmed strike or not}
• When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff {substitute in dagger here as it is a monk weapon}, you can also make an unarmed strike as a bonus action, assuming you haven’t already taken a bonus action this turn.