Is the nose of the solar system and the solar apex the same thing?
No, they're roughly 60 degrees apart.
The Solar Apex.
The solar apex does indeed indicate a component of direction of travel within our corner of the galaxy:
The coordinates as obtained by visual observation of the
apparent motion is right ascension 18h 28m 0s and declination
of 30° North in galactic coordinates: 56.24° longitude, 22.54°
latitude. The radioastronomical position is 18h 03m 50.2s and dec
30° 00′ 16.8″ galactic coordinates: 58.87° longitude, 17.72°
There are two different precise measurements here which conflict, I won't speculate as to why Wikipedia is unreferenced on the subject. The galaxy is a place containing a great deal of complex movement and the Local Standard of Rest is moving relative to the galactic plane, the centre of the galaxy and the other local stars:
In astronomy, the local standard of rest or LSR follows the mean
motion of material in the Milky Way in the neighborhood of the Sun.
The path of this material is not precisely circular. The Sun follows
the solar circle (eccentricity e < 0.1 ) at a speed of about 255 km/s
in a clockwise direction when viewed from the galactic north pole at a
radius of ≈ 8.34 kpc about the center of the galaxy, and
has only a slight motion, towards the solar apex, relative to the LSR.
We're traveling relative to an ever shifting cloud of stars towards Hercules (also shifting), southwest of the star Vega.
The sun's motion in the Milky Way is not confined to the galactic
plane; it also shifts ("bobs") up and down with respect to the plane
over millions of years.
So you see, there are many components of direction of travel to consider.
There's an unquotable abstract from a Harvard article here, which may give you an idea of how much needs to be taken into account in determining the relative vectors.
The ""Nose" of the solar system is sometimes used to refer to the direction of travel of the local galactic wind. Just as a comet's tail points away from the sun, the tail of the Heliopause points away from the wind, the nose points in the direction it's coming from.
The nose is as you have stated is pointing towards Scorpius.