It seems in today's world that to have conviction may cause someone to be labeled as a hateful person. Yet merely because someone has a conviction on an issue, does not indicate such a person is hateful.
In order to lay a good foundation, let us consider the Oxford definitions of "hate" and "conviction."
Here is the definition for "hate:"
Feel intense dislike for.
‘the boys hate each other’
‘he was particularly hated by the extreme right’
1.1 Have a strong aversion to (something)
‘he hates flying’
with infinitive ‘I'd hate to live there’
1.2 with infinitive Used politely to express one's regret or embarrassment at doing something.
‘I hate to bother you’
1.3 (hate on) informal no object Express strong dislike for; criticize or abuse.
‘I can't hate on them for trying something new’
So in the context of "hate speech," the definition used is that of an "intense dislike."
Now observe the definition of "conviction:"
1 A formal declaration by the verdict of a jury or the decision of a judge in a court of law that someone is guilty of a criminal offence.
‘she had a previous conviction for a similar offence’
2 A firmly held belief or opinion.
‘she takes pride in stating her political convictions’
[with clause] ‘his conviction that the death was no accident was stronger’
2.1 [mass noun] The quality of showing that one is firmly convinced of what one believes or says.
‘she had been speaking for some five minutes with force and conviction’
In regards to the context of hate speech vs. conviction, the second definition is used of "a firmly held belief or opinion." This definition yields a secondary question such that what does "firmly" mean?
So back to the Oxford dictionary we proceed in regards to the definition of "firmly:"
1 With little possibility of movement; securely.
‘the door remained firmly shut’
‘his work is rooted firmly in Chinese culture’
1.1 In a way that uses steady power or strength.
‘I had her wrist grasped firmly in my hand’
‘stop the bleeding by pressing firmly’
1.2 With resolute determination and strength of character.
‘the council is firmly committed to raising standards in education’
1.3 Without doubt; definitely.
‘I firmly believe that we could do better’
‘his reputation is now firmly established’
In regards to the context of "firmly" in regards to a firmly held belief or opinion, the definition used for "firmly" would most likely be, "With resolute determination and strength of character" and "Without doubt; definitely."
At this point we end up with conviction meaning a belief or opinion held with resolute determination, which could be without doubt and definitively. This is not an "intensive dislike" as defined by the term hate.
Let us now consider the definition of "intense" from hate's definition of an "intensive dislike:"
1 Of extreme force, degree, or strength.
‘the job demands intense concentration’
‘the heat was intense’
‘an intense blue’
1.1 (of an action) highly concentrated.
‘a phase of intense activity’
2 Having or showing strong feelings or opinions; extremely earnest or serious.
‘an intense young woman, passionate about her art’
‘a burning and intense look’
Here an "intensive dislike" becomes a dislike of extreme force, degree, or strength. Oxford further defines "extreme" as "Reaching a high or the highest degree; very great; outermost, utmost." So extreme is a measurement of degree or scale of preference in regards to something that is disliked.
So one can say that "I have a conviction that homosexuality is wrong" or that "I have a conviction that abortion is wrong" without speaking hatefully. Likewise, someone can also say that "I have a conviction that homosexuality is not wrong" or that "I have a conviction that abortion is not wrong" without speaking hatefully.
The factor that determines whether or not stating a conviction (on one side of an issue or another) is the degree to which emotion is used in the expression. For example, one could state, "I have a conviction that homosexuality is wrong because the Church Fathers taught against it, the Bible speaks against it, and it is contrary to the natural law of normal reproduction," by speaking plainly with resolute determination without speaking hatefully. Similarly someone could state, "I think abortion is okay because it's a woman's right to choose," by speaking with resolute determination without speaking hatefully. Someone can disagree with another's conviction by saying with resolute determination, "I don't think abortion is right because the value of life is more important that a person's right to choose," without speaking hatefully.
So someone merely stating a conviction about an issue does not, by default, indicate hate speak or that it is spoken hatefully.
Definitions acquired from the Oxford Dictionary Website at https://www.lexico.com/en.