Given: You have 60 numbers (1 through 60).  
Scenario 1.)What are the odds of selecting the following 6 numbers: 1,
2, 3, 4, 5, 6.
Scenario 2.)What are the odds of selecting the following 6 numbers: 1,
11, 17, 22, 34, 55.

Are the odds different and what are they?

The odds are the same for either group of six numbers or any group of
six numbers.  I assume you want to select all six of them with just
six picks without replacement.

The probability of selecting six given numbers without replacement is:

(6/60)*(5/59)*(4/58)*(3/57)*(2/56)*(1/55) = 1/50,063,860

Although mathematically the probability is the same, you just know 1,
2, 3, 4, 5, 6 is less likely to happen somehow.

This is where intuition fails you.  The set of numbers 1, 2, 3, 4, 5,
6 is not less likely to happen somehow than some other set.  But they
have a unique feel unlike any other set.  In reality you are comparing
by feel, this one special set against many other sets that all feel
similar to each other.  Not special.

1, 11, 17, 22, 34, 55
1, 12, 17, 22, 34, 55
1, 11, 17, 22, 34, 56
etc.

Three different sets.  But they all feel the same.  Not special.  They
all feel like NOT

1, 2, 3, 4, 5, 6

And so when one of them comes up, a set feels not special came up. 
And a not special set is more likely to come up, than a set that feels
special because there are so many more not special sets.  But a given,
individual not special set is no more likely to come up than the set
consisting of the first six counting numbers.

Of course, the real question asked is "what are the odds of SELECTING
the following 6 numbers: 1,2,3,4,5,6"

In the real lotery, I bet the numbers 1,2,3,4,5,6 are selected very
often by the people playing.  And because of this, if those number are
ever chosen as the winning numbers, then everyone who selected those
numbers will receive only a small portion of the total winnings
because they must split it with everyone else who selected those
numbers.